No, force is not equal between the shooter and shootee. Neither is energy. Only momentum.
Momentum isn't a "form of energy." Energy may as well be totally irrelevant. Kinetic energy is just force * distance, and all of the bullet's kinetic energy is turned into heat energy within a fraction of a second of the bullet coming to rest. Momentum, on the other hand, is transferred into the target, accelerating it a bit (not even enough to affect balance, of course). That's how physics works. 100% of the kinetic energy is turned into heat, and 100% of the momentum is conserved as momentum.
In the example given, the shooter gets hit with about 10 pounds of force, while the shootee gets 323 pounds. That's because of the enormous velocity difference between the gun and the bullet. The momentums are equal, but the forces, energies, and stresses are not.
There are a bunch of ways of figuring out the average force and stress, but using energy is the easiest.
Let's say you're shooting a 5" 1911. Within 5" (more like 4.5" really), the bullet is accelerated from 0 fps to let's say 828 fps, for 350 ft-lbs. 1 ft-lb is 1 pound of force over 1 foot of distance. 5" is 5/12 feet, so that's 840 pounds of force on the bullet. Bullet diameter is .451", so that's 5258 PSI on the bullet, on average.
Then the rest is the same.
Shooter: 10 lbs, 5 PSI
Shootee: 323 lbs, 840 PSI
Bullet: 840 lbs, 5258 PSI
(The fact that the PSI in the shootee and the force on the bullet are the same is just a coincidence.)
I'd say the bullet is getting the short end of the stick.
Remember, if we forget about air resistance, then the momentum of the bullet in flight, the momentum of the shooter and gun after firing, and the momentum of the shootee and bullet after getting hit, are all equal. Even though the forces are extremely different.
Also, high force or high stress does not correlate to high damage. The shallower the bullet penetrates, the higher the average force will be, because force = energy / distance (and also = mass * acceleration). So that means someone wearing a bulletproof vest will absorb a lot more force than someone with no armor.
Of course, this is all a gross oversimplification, especially reducing force to average force. In reality, in cases like this, force will be high at the start and gradually taper off (no, they don't start at the same number). Average force does give you something of an idea of the "real" force involved, though.
Oh, wait a minute, I did recoil wrong. It should be much higher. But still nowhere near what the bullet and shootee experience.